if(StartVoidLoopTime - LastTimeButtonWasPressed >= interval) //if the total time - the start time is greater than 3 sec

So what is 2 - 10 seconds? In unsigned subtraction, it's a really big number, which will most certainly be bigger than 3 seconds. You should be subtracting LastTimeButtonWasPressed from the current time, not some time you set an arbitrarily amount of time ago.

if (StartVoidLoopTime - LastTimeButtonWasPressed >= interval); //if the total time - the start time is greater than 3 sec That semicolon on the end forms the body of the if statement. A ; by itself is valid code - it is a no-op statement. Therefore, if the condition is true, nothing happens. There being no else statement with this if statement, nothing happens if the condition is not true. The whole point of the if statement - with that semicolon on the end - is what? It achieves nothing.

Servo myservo;const int leftButton = 2; //variable to represent the pin that the left button is onconst int rightButton = 4; //variable to represent the pin that the right button is onint caseNum = 0; //case number to tell the servo which case to runint pos = 0; //positionint delayTime = 20; // increasing this slows down the servo movementunsigned long lastPress = 0; // last time updatelong interval = 3000UL; // interval at which to do something (milliseconds)